Law of Exponents Cheatsheet

math exponents law

I was studying rationals this week so I thought that creating an exponents cheat sheet for future reference is a good idea.

Powers of 0

$$ a^0 = 1 $$

Example:

$$ 3^0 = 1 $$

Powers of 1

$$ a^1 = a $$

Example:

$$ 7^1 = 7 $$

Power of 1 with base 0

$$ 0^0 = Indeterminate $$

Multiplying powers with the same base

$$ a^{x}a^{y} = a^{x+y} $$
Example:
$$ \begin{equation} \begin{split} a^{2}a^{3} &= (aa)(aaa) \\ &= (aaaaa) \\ &= a^5 \end{split} \end{equation} $$
$$ \begin{equation} \begin{split} a^{2}a^{3} &= a^{2+3} \\ &= a^5 \end{split} \end{equation} $$

Dividing powers with the same base

$$ \frac{a^{x}}{a^{y}} = a^{x-y} $$

Example:

$$ \begin{equation} \begin{split} a^{5}a^{3} &= (aaaaa)/(aaa) \\ &= aa \\ &= a^2 \end{split} \end{equation} $$
$$ \begin{equation} \begin{split} a^{5}a^{3} &= a^{5-3} \\ &= a^2 \end{split} \end{equation} $$

Exponentiation of a power

$$ (a^{x})^y = a^{xy} $$

Example:

$$ \begin{equation} \begin{split} (a^2)^3 &= (aa)^3 \\ &= (aa)(aa)(aa) \\ &= (aaaaaa) \\ &= a^6 \end{split} \end{equation} $$
$$ \begin{equation} \begin{split} (a^2)^3 &= a^{2 \cdot 3} \\ &= a^6 \end{split} \end{equation} $$

Distributive Property of Exponents

$$ (ab)^x = a^{x}b^{x} $$

Example:

$$ (2 \cdot 3)^4 = 2^{4} \cdot 3^{4} = 16 \cdot 81 = 1296 $$

Distributive Property of Exponents on Fractions

$$ \Bigg(\frac{a}{b}\Bigg)^x = \frac{a^x}{b^x} $$

Example:

$$ \Bigg(\frac{7}{9}\Bigg)^4 = \frac{7^4}{9^4} = \frac{2401}{6561} $$

Negative One Property of Exponents

$$ a^{-n} = \frac{1}{a^n} $$

Example:

$$ 3^{-4} = \frac{1}{3^4} = \frac{1}{81} $$

Nth Root

$$ a^{\frac{x}{y}} = \sqrt[y]{a^x} $$

Example:

$$ 7^{\frac{2}{3}} = \sqrt[3]{7^2} = \sqrt[3]{49} $$